#9923645

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#55704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,5)(4,8)(6,9)(7,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,3,7,4)(2,8,10,5)(6,11,15,12)(9,14,16,13)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(35,43)(37,46)(39,48)(41,47);(18,23)(19,25)(24,31)(26,32)(33,35,38,43)(34,39,42,48)(36,37,44,46)(40,47,45,41);(1,4,7,3)(2,8,10,5)(6,12,15,11)(9,14,16,13)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,36,38,44)(34,40,42,45)(35,37,43,46)(39,47,48,41);(1,5,7,8)(2,3,10,4)(6,14,15,13)(9,12,16,11)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,37)(34,41)(35,44)(36,43)(38,46)(39,40)(42,47)(45,48);(1,6,7,15)(2,9,10,16)(3,11,4,12)(5,13,8,14)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,34)(35,39)(36,40)(37,47)(38,42)(41,46)(43,48)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830841);